Quaternionic Grover Walks and Zeta Functions of Graphs with Loops

Abstract

We define a quaternionic extension of the Szegedy walk on a graph and study its right spectral properties. The condition for the transition matrix of the quaternionic Szegedy walk on a graph to be quaternionic unitary is given. In order to derive the spectral mapping theorem for the quaternionic Szegedy walk, we derive a quaternionic extension of the determinant expression of the second weighted zeta function of a graph. Our main results determine explicitly all the right eigenvalues of the quaternionic Szegedy walk by using complex right eigenvalues of the corresponding doubly weighted matrix. We also show the way to obtain eigenvectors corresponding to right eigenvalues derived from those of doubly weighted matrix.Abbr. title: Quaternionic quantum walks of Szegedy type AMS 2010 subject classifications: 60F05, 05C50, 15A15, 11R52 quantum walk on a graph in this study. The discrete-time quantum walk in one dimension lattice was intensively studied by Ambainis et al. [4]. One of its most striking properties is the spreading property of the walker. The standard deviation of the position grows linearly in time, quadratically faster than classical random walk. On the other hand, a walker stays at the starting position, i.e., localization occurs. The reviews and books on quantum walks are Kempe [18], Konno [19], Venegas-Andraca [37], Cantero et al. [8], Manouchehri and Wang [27], Portugal [28] for examples.One of the most important discrete-time quantum walks on graphs is the Grover walk on a graph. The Grover walk on a graph was first formulated in [13] for complete graphs. After Grover's approach, several researchers, for example [38,3,39] proposed quantum walks on graphs. Emms et al. [9] and Godsil and Guo [12] applied the spectral property of the Grover walk to graph isomorphism problem. In the present article, we focus on the discrete-time quantum walk on a graph based on the quantum search algorithm proposed by Szegedy [39]. This walk arises naturally from a Markov chain, and is called the Szegedy walk.Zeta functions of graphs originate from the Ihara zeta function for a regular graph introduced by Ihara. In [17], Ihara defined a p-adic analogue of the Selberg zeta function associated to a certain kind of discrete subgroups of the 2 by 2 projective linear group over p-adic fields. After that, the Ihara zeta function has been extensively studied as a zeta function of a graph by many researchers [17,32,35,36,14,6,34,11,23]. Sato [31] defined a new zeta function (the second weighted zeta function) of a graph by modifying a determinant expression of the (first) weighted zeta function which had been proposed by Mizuno and Sato [25]. This new zeta function and its determinant formula play essential roles in the determination of eigenvalues of the quantum walk on a graph in Konno and Sato [22] and in another proof of the Smilansky's formula [33] for the characteristic polynomial of the bond scattering matrix of a graph in Mizuno and Sato [26]. Ren et al. [29] found an interesting relation between the Ihara zeta fun...

Probability Distributions and Weak Limit Theorems of Quaternionic Quantum Walks in One Dimension

“…The quaternionic quantum walk (QQW) on the one-dimensional lattice is introduced by Konno [1] as a natural quaternionic extension of QW. Konno, Mitsuhashi, and Sato [2,3,4] studied some properties about the spectrum of QQW on some graphs. Both QQWs are defined by extending complex components of the unitary matrix which governs the dynamics of corresponding QW to quaternion components.…”